Join-semidistributive lattices and convex geometries

We introduce the notion of a convex geometry extending the notion of a finite closure system with the anti-exchange property known in combinatorics. This notion becomes essential for the different embedding results in the class of join-semidistributive lattices. In particular, we prove that every finite join-semidistributive lattice can be embedded into a lattice S_P(A) of algebraic subsets of a suitable algebraic lattice A. This latter construction, S_P(A), is a key example of a convex geometry that plays an analogous role in hierarchy of join-semidistributive lattices as a lattice of equivalence relations does in the class of modular lattices. We give numerous examples of convex geometries that emerge in different branches of mathematics from geometry to graph theory. We also discuss the introduced notion of a strong convex geometry that might promise the development of rich structural theory of convex geometries.     

Uniform binary geometries

In a geometric lattice every interval can be mapped isomorphically into an upper interval (containing 1) by a strong map. A natural question thus arises as to what extent certain assumptions on the upper interval structure determine the whole lattice. We consider conditions of the following sort: that above a certain levelm any two upper intervals of the same length be isomorphic. This property, called uniformity, is studied for binary geometries. The geometries satisfying the strongest uniformity condition (m = 1) are determined (except for one open case). As is to be expected the corresponding problem for lower intervals is easier and is solved completely.     

On the greedy dimension of a partial order

This paper introduces a new concept of dimension for partially ordered sets. Dushnik and Miller in 1941 introduced the concept of dimension of a partial order P, as the minimum cardinality of a realizer, (i.e., a set of linear extensions of P whose intersection is P). Every poset has a greedy realizer (i.e., a realizer consisting of greedy linear extensions). We begin the study of the notion of greedy dimension of a poset and its relationship with the usual dimension by proving that equality holds for a wide class of posets including N-free posets, two-dimensional posets and distributive lattices.     

Projective mappings between projective lattice geometries

The concept of projective lattice geometry generalizes the classical synthetic concept of projective geometry, including projective geometry of modules.In this article we introduce and investigate certain structure preserving mappings between projective lattice geometries. Examples of these so-calledprojective mappings are given by isomorphisms and projections; furthermore all linear mappings between modules induce projective mappings between the corresponding projective geometries.     

Random orders of dimension 2

A relationship is established between (partially) ordered sets of dimension 2 chosen randomly on a labelled set, chosen randomly by isomorphism type, or generated by pairs of random linear orderings. As a consequence we are able to determine the limiting probability (in each of the above sample spaces) that a two-dimensional order is rigid, is uniquely realizable, or has uniquely orientable comparability graph; all these probabilities lie strictly between 0 and 1. Finally, we show that the number of 2-dimensional (partial) orderings of a labelled n-element set is .On leave from Emory University, Atlanta, GA. Research at Emory supported by ONR grant N00014 85-K-0769.     

Planar graphs and poset dimension

We view the incidence relation of a graph G=(V. E) as an order relation on its vertices and edges, i.e. a<_G b if and only of a is a vertex and b is an edge incident on a. This leads to the definition of the order-dimension of G as the minimum number of total orders on V E whose intersection is <_G. Our main result is the characterization of planar graphs as the graphs whose order-dimension does not exceed three. Strong versions of several known properties of planar graphs are implied by this characterization. These properties include: each planar graph has arboricity at most three and each planar graph has a plane embedding whose edges are straight line segments. A nice feature of this embedding is that the coordinates of the vertices have a purely combinatorial meaning.     

Connectedness and diameter for random orders of fixed dimension

Let P _k (n) be the (partial) order determined by intersecting k random linear orderings of a set of size n; equivalently, let P _k (n) consist of n points chosen randomly and independently from the unit cube in ^k , with the induced product order. We show for each fixed k>1, that with probability approaching 1 as n, the comparability graph of P _k (n) is connected and has diameter 3.     

Angle orders,regular n-gon orders and the crossing number

A finite poset P(X,<) on a set X={ x ₁,...,x _m} is an angle order (regular n-gon order) if the elements of P(X,<) can be mapped into a family of angular regions on the plane (a family of regular polygons with n sides and having parallel sides) such that x _ij if and only if the angular region (regular n-gon) for x _i is contained in the region (regular n-gon) for x _j. In this paper we prove that there are partial orders of dimension 6 with 64 elements which are not angle orders. The smallest partial order previously known not to be an angle order has 198 elements and has dimension 7. We also prove that partial orders of dimension 3 are representable using equilateral triangles with the same orientation. This results does not generalizes to higher dimensions. We will prove that there is a partial order of dimension 4 with 14 elements which is not a regular n-gon order regardless of the value of n. Finally, we prove that partial orders of dimension 3 are regular n-gon orders for n3.This research was supported by the Natural Sciences and Engineering Research Council of Canada, grant numbers A0977 and A2415.     

Circle orders,N-gon orders and the crossing number

Let ={P ₁,...,P _m } be a family of sets. A partial order P(, <) on is naturally defined by the condition P _i <P _j iff P _i is contained in P _j . When the elements of are disks (i.e. circles together with their interiors), P(, <) is called a circle order; if the elements of are n-polygons, P(, <) is called an n-gon order. In this paper we study circle orders and n-gon orders. The crossing number of a partial order introduced in [5] is studied here. We show that for every n, there are partial orders with crossing number n. We prove next that the crossing number of circle orders is at most 2 and that the crossing number of n-gon orders is at most 2n. We then produce for every n4 partial orders of dimension n which are not circle orders. Also for every n>3, we prove that there are partial orders of dimension 2n+2 which are not n-gon orders. Finally, we prove that every partial order of dimension 2n is an n-gon order.This research was supported under Natural Sciences and Engineering Research Council of Canada (NSERC Canada) grant numbers A2507 and A0977.     

The order dimension of multinomial lattices

Using the notion of Ferrers dimension of incidence structures, the order dimension of multi-nomial lattices (i.e. lattices of multi-permutations) is determined. In particular, it is shown that the lattice of all permutations on ann-element set has dimensionn–1.     

Largest extension of a finite convex geometry

We present a new embedding of a finite join-semidistributive lattice into a finite atomistic join-semidistributive lattice. This embedding turns out to be the largest extension, when applied to a finite convex geometry.In Celebration of the Sixtieth Birthday of Ralph N. McKenzieReceived September 18, 2002; accepted in final form September 29, 2003.     

Edge subdivision and dimension

The dimension of a partial order can be multiplied by an arbitrarily large factor when edges are subdivided.This research was supported by National Science Foundation grant DCR-8604577.     

Angle orders

A finite poset is an angle order if its points can be mapped into angular regions in the plane so thatx precedesy in the poset precisely when the region forx is properly included in the region fory. We show that all posets of dimension four or less are angle orders, all interval orders are angle orders, and that some angle orders must have an angular region less than 180° (or more than 180°). The latter result is used to prove that there are posets that are not angle orders.The smallest verified poset that is not an angle order has 198 points. We suspect that the minimum is around 30 points. Other open problems are noted, including whether there are dimension-5 posets that are not angle orders.Research supported in part by the National Science Foundation, grant number DMS-8401281.     

Residuated mappings

A survey is given of the ubiquitous rôle played by residuated mappings in the theory of ordered sets, lattices, and ordered semigroups. This is intended mainly for the young researcher interested in the general area of ordered sets. Some open questions that arise out of the presentation are listed in the final section.     

Cover-preserving order embeddings into Boolean lattices

It is not known which finite graphs occur as induced subgraphs of a hypercube. This is relevant in the theory of parallel computing. The ordered version of the problem is: Which finite posets P occur as cover-preserving subposets of a Boolean lattice? Our main Theorem gives (for 0,1-posets) a necessary and sufficient condition, which involves the chromatic number of a graph associated to P. It is applied respectively to upper balanced, meet extremal, meet semidistributive, and semidistributive lattices P. More specifically, we consider isometric embeddings of posets into Boolean lattices. In particular, answering a question of Ivan Rival to the positive, a nontrivial invariant for the covering graph of a poset is found.     

Chromatic number of Hasse diagrams,eyebrows and dimension

We construct posets of dimension 2 with highly chromatic Hasse diagrams. This solves a previous problem by Nesetril and Trotter.     

On the dimension of the cartesian product of relations and orders

It is known that for incidence structures and , max , wheref dim stands for Ferrers relation. We shall show that under additional assumptions on and , both bounds can be improved. Especially it will be shown that the square of a three-dimensional ordered set is at least four-dimensional.     

A note on the dimension of a poset

It is shown that the dimension of a poset is the smallest cardinal number such that there is an embedding of the poset into a strict product of linear orders.     

Ordered sets with no chains of ideals of a given type

We study the possible order types of chains of ideals in an ordered set. Our main result is this. Given an indecomposable countable order type , there is a finite listA ₁ , ...,A _n of ordered sets such that for every ordered setP the setJ(P) of ideals ofP, ordered by inclusion, contains a chain of type if and only ifP contains a subset isomorphic to one of theA ₁ ^#x03B1; , ...,A _n . The finiteness of the list relies on the notion of better quasi-ordering introduced by Nash-Williams and the properties of scattered chains obtained by Laver.The results presented. here constitute the second chapter of the third cycle thesis presented by the second author before the Claude Bernard University, Lyon (July 1983).     

Morphisms of projective geometries and semilinear maps

Morphisms between projective geometries are introduced; they are partially defined maps satisfying natural geometric conditions. It is shown that in the arguesian case the morphisms are exactly those maps which in terms of homogeneous coordinates are described by semilinear maps. If one restricts the considerations to automorphisms (collineations) one recovers the so-called fundamental theorem of projective geometry, cf. Theorem 2.26 in [2].Supported by a grant from the Fonds National Suisse de la Recherche Scientifique.